Prove that in the two-party supply chain discussed in class ie a manufacture and a retailer a buy-back contract can be viewed as a special type of options contracts Hint For a given buy back contract

A buy-back contract is an agreement between a manufacturer and a retailer where the manufacturer agrees to buy back any unsold products from the retailer at a pre-determined price. This type of contract can be viewed as a special type of options contract.

To see this, consider the following example. Suppose a manufacturer and a retailer enter into a buy-back contract for a certain product. The contract specifies that the manufacturer will buy back any unsold products from the retailer at a price of $10 per unit. Let's assume that the retailer initially purchases 100 units of the product from the manufacturer at a cost of $8 per unit.

Now, let's construct an options contract with the following parameters:

• Strike price: $10
• Premium: $0 (i.e., the retailer does not pay any upfront cost to purchase the option)
• Expiration date: The same as the end of the selling season, which is when the buy-back contract expires.

Under this options contract, the retailer has the right, but not the obligation, to sell any unsold products back to the manufacturer at a price of $10 per unit. If the retailer sells any units back to the manufacturer, it will receive a profit of $2 per unit ($10 buy-back price minus $8 initial cost).

Now, let's consider the sales and costs under both the buy-back contract and the options contract. If the retailer sells all 100 units of the product, it will make a profit of $200 under both contracts. If the retailer sells only 50 units and has to sell the remaining 50 units back to the manufacturer, it will make a profit of $100 under both contracts. If the retailer sells all 100 units back to the manufacturer, it will break even under both contracts.

Therefore, we can see that the sales as well as the costs are the same under both contracts. This means that a buy-back contract can be viewed as a special type of options contract with a strike price equal to the buy-back price and no upfront cost to purchase the option.